Question: Given the equation: $ y = -4x^2 + 24x - 31$ Find the parabola's vertex.
Solution: When the equation is rewritten in vertex form like this, the vertex is the point $({h}, {k})$ $ y = A(x - {h})^2 + {k} $ We can rewrite the equation in vertex form by completing the square. First, move the constant term to the left side of the equation: $ \begin{eqnarray} y &=& -4x^2 + 24x - 31 \\ \\ y + 31 &=& -4x^2 + 24x \end{eqnarray} $ Next, we can factor out a $-4$ from the right side: $ y + 31 = -4(x^2 - 6x) $ We can complete the square by taking half of the coefficient of our $x$ term, squaring it, and adding it to both sides of the equation. The coefficient of our $x$ term is $-6$ , so half of it would be $-3$ , and squaring that gives us ${9}$ . Because we're adding the $9$ inside the parentheses on the right where it's being multiplied by $-4$ , we need to add ${-36}$ to the left side to make sure we're adding the same thing to both sides. $ \begin{eqnarray} y + 31 &=& -4(x^2 - 6x) \\ \\ y + 31 + {-36} &=& -4(x^2 - 6x + {9}) \\ \\ y - 5 &=& -4(x^2 - 6x + 9) \end{eqnarray} $ Now we can rewrite the expression in parentheses as a squared term: $ y - 5 = -4(x - 3)^2 $ Move the constant term to the right side of the equation. Now the equation is in vertex form: $ y = -4(x - 3)^2 + 5 $ Now that the equation is written in vertex form, the vertex is the point $({h}, {k})$ $ y = A(x - {h})^2 + {k} $ $ y = -4(x - {(3)})^2 + {(5)} $ The vertex is $({3}, {5})$. Be sure to pay attention to the signs when interpreting an equation in vertex form.